MR. IVORY ON THE THEORY OP ASTRONOMICAL REFRACTIONS. 223 



zenith ; and they can be computed in no other way than by reducing them to series 

 of the powers of e. By substituting for "k Q^, the equivalent series already known, we 

 immediately obtain 



Further, by expanding S and its differential, the expression of T will take this form, 

 T = sin«-?^^ii4~ -{6363 + G,e' + G<,e9 + &c. } ; 



and we shall have 



G3 = Ai — A3 + 2 B3 = 0-2436 



G5 = — Ai + 3 A3 — 2 A5 + 2 B5 = 0-4523 



G7 = Ai — 3 A3 + 5 A5 — 3 A; + 2 B7 = 0-4705 



G9 = - Ai + 3 A3 - 5 A5 + 7 A; — 4 Ag + 2 By = 0-3502 



C^ii = Ai — 3 A3 + 5 A5 — 7 A7 + 9 Ac, — 5 All + 2 Bu = 0-2092 



Gi3 = - Aj + 3 A3 — 5 A5 + 7 A; - 9 Ag 4- 1 1 All - 6 Ai3 + 2 Bi3 = 0*1050. 



The series for T and b being now known, the coefficients of the terms must next be 

 expressed in seconds of a degree ; which being done, the following final results will 

 be obtained. 



The values of T and h are added in separate columns of the annexed table for alti- 

 tudes less than 10°: for greater altitudes they are omitted as of no account. The 

 application for finding the corrected refraction from the formula 



will best be explained by the examples afterwards given. 



14. The Theoretical Table of refractions which has been computed by the foregoing 

 formulas, and which is deduced solely from the phenomena of the atmosphere without 

 arbitrary assumptions, is next to be compared with the tables most esteemed by astro- 

 nomers. Two tables more eminently deserve this character ; namely, Bessel's table 

 with its supplement in the Tahulce Regiomontanoe, which may be considered as the 

 result of observations, and as being nearly exact to 88° or 88°J from the zenith ; and 



